There are 12 Trophies that can be collected.

BRONZE

Mad with Power
Down an opponent with a Super Combo.

This is Where You Fall Down
Down an opponent without using a Super Combo or Tandem Attack.

Crash and Burn
Down an opponent while they are Stand Crashed.

The Kid Gloves are Off
Down an opponent while in child form.

Strong and Attractive
Raise Mahrahia's magnetic power to maximum level.

Out of the Cage
Win one Ranked Match online.

Show and Tell
Upload replay data.

SILVER

Stardust Crusaders
Win a fight with 22 different characters.

Patty Cake, Patty Cake
Activate a Blazing Fists match.

The World is Your Oyster
Counter a time-freeze.

Never Saw it Coming
K.O. N'Dool via Perfect in bonus stage.

GOLD

Standing the Test of Time
Complete Story Mode or Challenge Mode for any character on any difficulty without using continues.

http://www.ps3trophies.org/game/jojos-bizarre-adventure-hd/trophies/